A000995 -id:A000995 - cp's OEIS frontend

A351132 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5x)) / (1 - 5x).

0, 1, 0, 1, 10, 76, 530, 3701, 27810, 237151, 2316350, 25135126, 292106400, 3559029501, 45211131460, 600619791201, 8384107777030, 123237338584576, 1904128564485610, 30789744821412401, 518479182191232950, 9057086806410632751, 163745788914416588050

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

Shifts 2 places left under 5th-order binomial transform.

Cf. A000995, A005011, A351028, A351053, A351056, A351128, A351151, A351161.

nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6x)) / (1 - 6x).

Original entry on oeis.org

0, 1, 0, 1, 12, 109, 900, 7309, 62280, 590185, 6402360, 78347593, 1042633908, 14648616757, 214421295132, 3266839420021, 52041902492496, 870810496011793, 15326196662766384, 283049655668743249, 5460180803581446684, 109489002283248831037, 2273856664328893182324

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

Shifts 2 places left under 6th-order binomial transform.

Cf. A000995, A005012, A351028, A351053, A351057, A351128, A351132, A351152.

nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).

A246118 T(n,k), for n,k >= 1, is the number of partitions of the set [n] into k blocks, where, if the blocks are arranged in order of their minimal element, the odd-indexed blocks are all singletons.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 11, 6, 1, 0, 1, 5, 26, 23, 9, 1, 0, 1, 6, 57, 72, 50, 12, 1, 0, 1, 7, 120, 201, 222, 86, 16, 1, 0, 1, 8, 247, 522, 867, 480, 150, 20, 1, 0, 1, 9, 502, 1291, 3123, 2307, 1080, 230, 25, 1, 0, 1, 10, 1013, 3084, 10660, 10044, 6627, 2000, 355, 30, 1

Offset: 1

Peter Bala, Aug 14 2014

Unsigned matrix inverse of A246117. Analog of the Stirling numbers of the second kind, A048993.
This is the triangle of connection constants between the monomial polynomials x^n and the polynomial sequence [x, x^2, x^2*(x - 1), x^2*(x - 1)^2, x^2*(x - 1)^2*(x - 2), x^2*(x - 1)^2*(x - 2)^2, ...]. An example is given below.
Except for differences in offset, this triangle is the Galton array G(floor(k/2),1) in the notation of Neuwirth with inverse array G(-floor(n/2),1).
Essentially the same as A256161. - Peter Bala, Apr 14 2018
From Peter Bala, Feb 10 2020: (Start)
The sums S(n):= Sum_{k >= 0} k^n*(x^k/k!)^2, n = 2,3,4,..., can be expressed as a linear combination of the sums S(0) and S(1) with polynomial coefficients, namely, S(n) = E(n,x)*S(0) + (1/x)*O(n,x)* S(1,x), where E(n,x) = Sum_{k >= 1} T(n,2*k)*x^(2*k) and O(n,x) = Sum_{k >= 0} T(n,2*k+1)*x^(2*k+1) are the even and odd parts of the n-th row polynomial of this array. This result is the analog of the Dobinski formula Sum_{k >= 0} (k^n)*x^k/k! = exp(x)*Bell(n,x), where Bell(n,x) is the n-th row polynomial of A048993.
For example, for n = 6 we have S(6) = Sum_{k >= 1} k^6*(x^k/k!)^2 = (x^2 + 11*x^4 + x^6) * Sum_{k >= 0} (x^k/k!)^2 + (1/x)*(4*x^3 + 6*x^5) * Sum_{k >= 1} k*(x^k/k!)^2.
Setting x = 1 in the above result gives Sum_{k >= 0} k^n*/k!^2 = A000994(n)*Sum_{k >= 0} 1/k!^2 + A000995(n)*Sum_{k >= 1} k/k!^2. See A086880. (End)

			Triangle begins
n\k| 1    2    3    4    5    6    7    8
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    2    1
5  | 0    1    3    4    1
6  | 0    1    4   11    6    1
7  | 0    1    5   26   23    9    1
8  | 0    1    6   57   72   50   12    1
...
Connection constants: Row 6 = (0, 1, 4, 11, 6, 1) so
x^6 = x^2 + 4*x^2*(x - 1) + 11*x^2*(x - 1)^2 + 6*x^2*(x - 1)^2*(x - 2) + x^2*(x - 1)^2*(x - 2)^2.
Row 5 = [0, 1, 3, 4, 1]. There are 9 set partitions of {1,2,3,4,5} of the type described in the Name section:
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Number of      Set partitions                Count
blocks
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
2                {1}{2,3,4,5}                   1
3           {1}{2,4,5}{3}, {1}{2,3,5}{4},
            {1}{2,3,4}{5}                       3
4          {1}{2,3}{4}{5}, {1}{2,4}{3}{5},
           {1}{2,5}{3}{4}, {1}{2}{3}{4,5}       4
5          {1}{2}{3}{4}{5}                      1

Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.
Emrah Kiliç and Helmut Prodinger, Identities with Squares of Binomial Coefficients: an Elementary and Explicit Approach, Publications de l'Institut Mathématique (Beograd) (N.S.), Vol.99(113) (2016), 243-248. See p. 248.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Tech Report TR 99-05, 1999.
E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.

Cf. A000295 (column 4), A007476 (row sums), A008277, A045618 (column 5), A048993, A246117 (unsigned matrix inverse), A256161, A000994, A000995, A086880.

Flatten[Table[Table[Sum[StirlingS2[j,Floor[k/2]] * StirlingS2[n-j-1,Floor[(k-1)/2]],{j,0,n-1}],{k,1,n}],{n,1,12}]] (* Vaclav Kotesovec, Feb 09 2015 *)

T(n,k) = Sum_{i = 0..n-1} Stirling2(i, floor(k/2))*Stirling2(n-i-1, floor((k - 1)/2)) for n,k >= 1.
Recurrence equation: T(1,1) = 1, T(n,1) = 0 for n >= 2; T(n,k) = 0 for k > n; otherwise T(n,k) = floor(k/2)*T(n-1,k) + T(n-1,k-1).
O.g.f. (with an extra 1): A(z) = 1 + Sum_{k >= 1} (x*z)^k/( ( Product_{i = 1..floor((k-1)/2)} (1 - i*z) ) * ( Product_{i = 1..floor(k/2)} (1 - i*z) ) ) = 1 + x*z + x^2*z^2 + (x^2 + x^3)*z^3 + (x^2 + 2*x^3 + x^4)*z^4 + .... satisfies A(z) = 1 + x*z + x^2*z^2/(1 - z)*A(z/(1 - z)).
k-th column generating function z^k/( ( Product_{i = 1..floor((k-1)/2)} (1 - i*z) ) * ( Product_{i = 1..floor(k/2)} (1 - i*z) ) ).
Recurrence for row polynomials: R(n,x) = x^2*Sum_{k = 0..n-2} binomial(n-2,k)*R(k,x) with initial conditions R(0,x) = 1 and R(1,x) = x. Compare with the recurrence satisfied by the Bell polynomials: Bell(n,x) = x*Sum_{k = 0..n-1} binomial(n-1,k) * Bell(k,x).
Row sums are A007476.

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 6, 11, 23, 60, 179, 553, 1716, 5415, 17801, 61956, 228391, 882309, 3530322, 14531621, 61454091, 267479778, 1200680113, 5561767211, 26553471186, 130366882251, 656668581417, 3387887246292, 17886582294921, 96603394562849, 533645344137390, 3014295344076655

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..695

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346051, A346052.

nmax = 31; A[] = 0; Do[A[x] = x + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346050
    if (n<3): return (0,1,1)[n]
    else: return sum(binomial(n-3,k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 28 2022

a(0) = 0, a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k).

A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 5, 12, 28, 68, 181, 531, 1671, 5491, 18627, 65299, 237880, 903907, 3580619, 14729777, 62639952, 274442521, 1236730244, 5729809348, 27292248240, 133614280479, 671803041553, 3464970976743, 18309428363425, 99010800275743, 547462187824465, 3093329527120022

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..695

Cf. A000994, A000995, A000996, A000997, A000998.

Cf. A007476, A210540, A346050, A346052.

function a(n)
  if n lt 3 then return (1+(-1)^n)/2;
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

nmax = 31; A[] = 0; Do[A[x] = 1 + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346051
    if (n<3): return (1, 0, 1)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

a(0) = 1, a(1) = 0, a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k).

A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 5, 11, 29, 80, 222, 630, 1881, 6004, 20420, 72979, 270659, 1035590, 4087205, 16675630, 70440641, 307933393, 1390117953, 6462787357, 30871458702, 151298796000, 760250325004, 3915477534861, 20662363081756, 111662169790416, 617482470676567, 3490973387652861

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..650

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346050, A346051.

function a(n) // a = A346052
  if n lt 3 then return Floor((3-n)/2);
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346052
    if (n<3): return (1, 1, 0)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

a(0) = a(1) = 1, a(2) = 0; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k).

A345178 a(0) = 0, a(1) = 1; a(n+2) = Sum_{k=0..n} Stirling2(n,k) * a(k).

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 8, 38, 194, 1138, 8154, 71544, 739406, 8674238, 113451160, 1648133190, 26631054962, 478633871152, 9531297220728, 208851860234540, 4997665703050398, 129765874491438094, 3639593254921626678, 109942671192206473592, 3569449102675488493032, 124319448405579907085938

Offset: 0

Ilya Gutkovskiy, Jun 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..355

Cf. A000995, A003659, A007469, A345177.

b:= proc(n, m) option remember; `if`(n=0,
      a(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> `if`(n<2, n, b(n-2, 0)):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 13 2021

a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = x + Normal[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

A137854 Triangle generated from an array: A008277 * A008277(transform).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 11, 8, 1, 1, 16, 28, 28, 16, 1, 1, 32, 71, 87, 71, 32, 1, 1, 64, 184, 266, 266, 184, 64, 1, 1, 128, 491, 823, 952, 823, 491, 128, 1, 1, 256, 1348, 2598, 3381, 381, 2598, 1348, 2561

Offset: 1

Gary W. Adamson, Feb 15 2008

Row sums = A000995 such that row 1 = A000995(3) = 1.
This array is the product of the lower triangular Stirling matrix and its transpose, which explains why the array is symmetric. - David Callan, Dec 02 2011
In the triangle, T(n,k) is the number of permutations of [n+1] that avoid both dashed patterns 1-23 and 3-12, start with an ascent, and have first entry k. For example, T(4,2)=4 counts 23154, 24153, 24315, 25431. - David Callan, Dec 02 2011

			First few rows of the array:
  1,  1,  1,   1,   1,    1, ...
  1,  2,  4,   8,  16,   32, ...
  1,  4, 11,  28,  71,  184, ...
  1,  8, 28,  87, 266,  823, ...
  1, 16, 71, 266, 952, 3381, ...
  ...
First few rows of the triangle:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   8,  11,   8,   1;
  1,  16,  28,  28,  16,   1;
  1,  32,  71,  87,  71,  32,   1;
  1,  64, 184, 266, 266, 184,  64,   1;
  1, 128, 491, 823, 952, 823, 491, 128,   1;
  ...

Cf. A000995, A008277.

Triangle read by rows = antidiagonals of an array formed by A008277 * A008277(transform), where A008277 = the Stirling number of the second kind triangle.

A346079 G.f. A(x) satisfies: A(x) = x - x^2 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

0, 1, 0, -1, -2, -2, 2, 17, 54, 109, 54, -796, -5000, -19499, -52252, -44617, 577554, 5071906, 27330978, 108557573, 263947354, -453137963, -11252508862, -92193933208, -545809325184, -2441788385255, -6271647457176, 22814756330975, 492197181810550, 4609129908957190

Offset: 0

Ilya Gutkovskiy, Jul 04 2021

sign

Cf. A000587, A000995, A336970, A346078.

nmax = 29; A[] = 0; Do[A[x] = x - x^2 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = -Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 29}]

a(0) = 0, a(1) = 1; a(n) = -Sum_{k=0..n-2} binomial(n-2,k) * a(k).

A110855 Table T(n,k), n >= 0, k >= 0, product M*M^(T) where M is the lower triangular matrix in A048993 (Stirling2 numbers) and M^(T) denotes the transpose matrix of M, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 8, 11, 8, 1, 0, 0, 1, 16, 28, 28, 16, 1, 0, 0, 1, 32, 71, 87, 71, 32, 1, 0, 0, 1, 64, 184, 266, 266, 184, 64, 1, 0, 0, 1, 128, 491, 823, 952, 823, 491, 128, 1, 0, 0, 1, 256, 1348, 2598, 3381, 3381

Offset: 0

Philippe Deléham, Sep 17 2005

			Matrix M:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  0, 1, 0, 0, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, 0, 0, ...
  0, 1, 3, 1, 0, 0, 0, 0, ...
  0, 1, 7, 6, 1, 0, 0, 0, ...
  ...
Matrix M^(T):
  1, 0, 0, 0, 0,  0, ...
  0, 1, 1, 1, 1,  1, ...
  0, 0, 1, 3, 7, 15, ...
  0, 0, 0, 1, 6, 25, ...
  0, 0, 0, 0, 1, 10, ...
  0, 0, 0, 0, 0,  1, ...
  ...
Table begins:
  1, 0,   0,   0,   0,   0,   0,   0,   0, 0, 0, ...
  0, 1,   1,   1,   1,   1,   1,   1,   1, 1,
  0, 1,   2,   4,   8,  16,  32,  64, 128, ...
  0, 1,   4,  11,  28,  71, 184, 491, ...
  0, 1,   8,  28,  87, 266, 823, ...
  0, 1,  16,  71, 266, 952, ...
  0, 1,  32, 184, 823, ...
  0, 1,  64, 491, ...
  0, 1, 128, ...
  0, 1, ...
  0, ...

Diagonal sums: 1, 0, 1, 2, 4, 10, 29, 90, 295, ... see A000995.

Main diagonal: 1, 1, 2, 11, 87, 952, 13513, ... see A047797.

cp's OEIS Frontend

A351132 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

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A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

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A246118 T(n,k), for n,k >= 1, is the number of partitions of the set [n] into k blocks, where, if the blocks are arranged in order of their minimal element, the odd-indexed blocks are all singletons.

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A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

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A345178 a(0) = 0, a(1) = 1; a(n+2) = Sum_{k=0..n} Stirling2(n,k) * a(k).

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A137854 Triangle generated from an array: A008277 * A008277(transform).

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A346079 G.f. A(x) satisfies: A(x) = x - x^2 * A(x/(1 - x)) / (1 - x).

A351132 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5x)) / (1 - 5x).

A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6x)) / (1 - 6x).